Dasgupta, S and Kakde, M (2024) BRUMER�STARK UNITS AND EXPLICIT CLASS FIELD THEORY. In: Duke Mathematical Journal, 173 (8). pp. 1477-1555.
Full text not available from this repository.Abstract
Let F be a totally real field of degree n, and let p be an odd prime. We prove the p-part of the integral Gross�Stark conjecture for the Brumer�Stark p-units living in CM abelian extensions of F . In previous work, the first author showed that such a result implies an exact p-adic analytic formula for these Brumer�Stark units up to a bounded root of unity error, including a �real multiplication� analogue of Shimura�s celebrated reciprocity law from the theory of complex multiplication. In this paper, we show that the Brumer�Stark units, along with n - 1 other easily described elements (these are simply square roots of certain elements of F ) generate the maximal abelian extension of F . We therefore obtain an unconditional construction of the maximal abelian extension of any totally real field, albeit one that involves p-adic integration for infinitely many primes p. Our method of proof of the integral Gross�Stark conjecture is a generalization of our previous work on the Brumer�Stark conjecture. We apply Ribet�s method in the context of group ring valued Hilbert modular forms. A key new construction here is the definition of a Galois module rL that incorporates an integral version of the Greenberg�Stevens L-invariant into the theory of Ritter�Weiss modules. This allows for the reinterpretation of Gross�s conjecture as the vanishing of the Fitting ideal of rL. This vanishing is obtained by constructing a quotient of rL whose Fitting ideal vanishes using the Galois representations associated to cuspidal Hilbert modular forms. © 2024 Duke University Press. All rights reserved.
| Item Type: | Journal Article |
|---|---|
| Publication: | Duke Mathematical Journal |
| Publisher: | Duke University Press |
| Additional Information: | The copyright for this article belongs to the publisher. |
| Department/Centre: | Division of Physical & Mathematical Sciences > Mathematics |
| Date Deposited: | 01 Aug 2024 06:47 |
| Last Modified: | 01 Aug 2024 06:47 |
| URI: | http://eprints.iisc.ac.in/id/eprint/85756 |
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